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Dynamic Grasping for an Arbitrary Polyhedral Object
by a Multi-Fingered Hand-Arm System
Akihiro Kawamura, Kenji Tahara, Ryo Kurazume and Tsutomu Hasegawa
Abstract— This paper proposes a novel control method forstable grasping using a multi-fingered hand-arm system withsoft hemispherical finger tips. The proposed method is simplebut easily achieves stable grasping of an arbitrary polyhe-dral object using an arbitrary number of fingers. Firstly, weformulate nonholonomic constraints between a multi-fingeredhand-arm system and an object constrained by rolling contactwith finger tips, and derive a condition for stable grasping bystability analysis. A new index for evaluating the possibility ofstable grasping is proposed and efficient initial relative positionsbetween finger tips and the object for realizing stable grasping
are analyzed. The stability of the proposed system and thevalidity of the index are verified through numerical simulations.
I. INTRODUCTION
A multi-fingered hand-arm system has been expected to
realize dexterous grasping like a human hand. Robots with
this system will be able to perform various manipulation
tasks even in an unstructured environment safely. Many
robotics systems and control methods for grasping an object
with an arbitrary shape have been proposed [1–5]. Especially,
dynamic grasping controllers using rolling constraints have
been reported [6–9]. However, these controllers are based on
inverse dynamics calculation, and object information such
as the mass and the shape of the object is required. In
contrast, there are several researches for dynamic grasping of
an unknown object using external sensing devices such as a
vision sensor and a tactile sensor [10–12]. In these methods,
sensing devices are costly and sensing error must be taken
into consideration though object information is not required
in advance.
On the other hand, Wimbock et al. [13] proposed a dy-
namic grasping method for an arbitrary object which requires
neither object information nor external sensors. However,
the stability of the system and the convergence performance
of the closed-loop dynamics were not discussed. Moreover,
rolling constraints between fingertips and the object surfaces
are not considered in their method. Similarly, Arimoto et al.
[14–17] have proposed a dynamic object grasping method
without object information and external sensors. The object
This work was partially supported by ”the Kyushu University ResearchSuperstar Program (SSP)”, based on the budget of Kyushu Universityallocated under President’s initiative.
A. Kawamura is with the Graduate School of Information Science andElectrical Engineering, Kyushu University, Fukuoka, 819-0395, [email protected]
K. Tahara is with the Organization for the Promotion of Ad-vanced Research, Kyushu University, Fukuoka, 819-0395, [email protected]
R. Kurazume and T. Hasegawa are with the Faculty of InformationScience and Electrical Engineering, Kyushu University, Fukuoka, JAPANkurazume, [email protected]
3aq
4aq
5aq
11q12q
13q
21q
22q
23q
31q32q
y
y&
1aq
2aq33q
xz
O
01x03x
02x
x&z&
Fig. 1. Multi-fingered hand-arm system
is limited to the one which consists of two flat and parallel
surfaces, though they verified the stability of the system and
the convergent performance of the closed-loop dynamics.
This paper proposes a novel control method for stable
grasping of an arbitrary polyhedral object using a multi-
fingered hand-arm system with soft hemispherical finger
tips. The proposed method is simple but easily achieves
stable grasping of an object without use of preliminary
information for a grasped object and any external sensors
using an arbitrary number of fingers. Firstly, we formulate
nonholonomic constraints between a multi-fingered hand-
arm system and an object constrained by rolling contact with
finger tips. The nonholonomic constraints for rolling contact
were proposed by Arimoto et al. [14–17] for an object with
two flat and parallel surfaces. We expand these constraints
for an object with an arbitrary polyhedral shape. The number
of fingers is also variable in our formulation. Secondly,
we derive Lagrange’s equation of motion for a hand-arm
system, and propose a new control signal which achieves
stable grasping. A new index for evaluating the possibility
of stable grasping is proposed and efficient initial relative
positions between finger tips and an object for realizing
stable grasping are analyzed. Using this index, it is clarified
that appropriate initial positions of finger tips depend on the
shape of the object and the ratio of the size of the object
and the radius of the finger tip. The stability of the proposed
system and the validity of the index are verified through
numerical simulations.
The 2009 IEEE/RSJ International Conference onIntelligent Robots and SystemsOctober 11-15, 2009 St. Louis, USA
978-1-4244-3804-4/09/$25.00 ©2009 IEEE 2264
II. A MULTI FINGERED HAND ARM SYSTEM
In this section, we define a model of a hand-arm system
composed of an arm and a multi-fingered hand. An example
of a multi-fingered hand-arm system treated here is illus-
trated in Fig. 1. An object to be grasped is an arbitrary
polyhedral object whose surfaces touched by finger tips are
flat. All finger tips maintain rolling contact with the object
surfaces, and do not slip and detach from the surfaces during
movement of the tips. Assume that fingertips roll within the
ranges of hemisphere surfaces, and they don’t deviate from
each contact surface. Note that the gravity effect is ignored in
this paper in order to have a physical insight into analyzing
physical interaction and stability of the system. As shown
in Fig. 1, O denotes the origin of Cartesian coordinates.
x0i ∈ R3 is the center of each contact area. Hereafter, the
subscript of i refers to the ith finger in all equations. The
degrees of freedom of the arm and the ith finger are Na
and Ni, respectively. The joint angle of the arm is expressed
by qa ∈ RNa . Similarly, the joint angle of the ith finger
is expressed by q0i ∈ RNi . q denotes the joint angles of
the arm and all the fingers(
= (qa, q01, q02, ..., q0N )T)
. N
is the number of the fingers. As shown in Fig. 2, Oc.m.
denotes the center of the object mass and the origin of local
coordinates. Its position in Cartesian coordinates is expressed
as x = (x, y, z)T∈ R
3. Instantaneous rotational axis of the
object at Oc.m. is expressed by ω. The orientation angular
velocities around each axis of Cartesian coordinates x, y, z
are expressed as ωx, ωy, ωz respectively.
A. Constraints
3-dimentional rolling constraints with area contacts are
modeled here. The orientation of the object in Cartesian
coordinates can be expressed by the rotational matrix R such
that
Rob = (rX , rY , rZ) ∈ SO (3) , (1)
where rX , rY , rZ ∈ R3 are mutually orthonormal vectors
on the object frame. It is known that this rotational matrix
is one of the members of the group SO(3). In addition to
this, we define contact frames at the center of each contact
area as follows:
RobRCi = (CiX ,CiY ,CiZ) , (2)
where RCi is the rotational matrix between the object frame
to the contact frames. Now, we consider the velocity of the
center of the contact area vi must satisfy
vi = ∆ri
(
CiY − ωi ×CiY
)
, (3)
where ωi ∈ R3 is the orientation angular velocity vectors for
each robotic finger on the contact frames, ri is the radius of
each finger tip, and ∆ri are the distance between the center
of the finger tips and the contact surfaces (see Fig. 2). vi
is on the tangential plane at the center of the contact area
(now, they are surfaces of the object). The rolling constraints
ix
∆ iriXC
iYC
iZC
Yr&
i&
iv ix0
Contact areaCenter of
contact area
Object
Finger tip
Center of
finger tip
Z
..mcOZr
XrY
iX
iY
ivir
Center of the
Object mass
iZ
Fig. 2. Contact model at the center of the contact area
are expressed such that the velocity of the center of the
contact area on the finger tip, given as (3), should equal
to the velocity of the center of the contact area on the object
surface
∆riCTiX
(
CiY − ωi ×CiY
)
= Xi (4)
∆riCTiZ
(
CiY − ωi ×CiY
)
= Zi, (5)
where
Xi = −C⊤
iX(x− x0i) (6)
Zi = −C⊤
iZ(x− x0i). (7)
Equations (4) and (5) denote nonholonomic rolling con-
straints on the object surfaces. These constrains are of linear
with respect to each velocity vector, and thereby they can be
expressed as Pfaffian constraints as follows:
Xiqq +Xixx+X iωω = 0 (8)
Ziq q +Zixx+Ziωω = 0, (9)
where
Xiq = ∆riCTiZJΩi −CT
iXJ0i
Xix = CTiX
Xiω = CiX × (x− x0i)T −∆riC
TiZ
Ziq = −∆riCTiXJΩi −CT
iZJ0i
Zix = CTiZ
Ziω = CiZ × (x− x0i)T+∆riC
TiX ,
(10)
and ω = (ωx, ωy, ωz)T
∈ R3 is the angular velocity
vector for the instantaneous rotational axis of the object.
JΩi∈ R
3×(Na+∑
N
i=1Ni) is the Jacobian matrix for the
orientation angular velocities of the finger tips with respect
2265
to the joint angular velocity q ∈ RNa+
∑N
i=1Ni . J0i ∈
R3×(Na+
∑N
i=1Ni) is the Jacobian matrix for the center of
the finger tip x0i with respect to the joint angle, respectively.
B. Contact Model of Soft Finger-Tip
In this paper, the physical relationship between the de-
formation of the finger tip at the center of the contact area
and its reproducing force is given on the basis of lumped-
parameterized model proposed by Arimoto et al [14]. The
reproducing force f (∆r) in the normal direction to the
object surface at the center of the contact area is given as
follows:[
fi = fi + ξiddt(ri −∆ri)
fi = k(ri −∆ri)2,
(11)
where k is a positive stiffness constant which depends on the
material of the finger tip. In the second term of the right-
hand side of the upper equation of (11), ξi (∆ri) is a positive
scalar function with respect to ∆ri, and thereby the viscous
force increases according to the expansion of the contact
area.
C. Overall Dynamics
The total kinetic energy for the overall system can be
described as follows:
K =1
2qTHq +
1
2xTMx+
1
2ωTIω, (12)
where H ∈ R(Na+
∑N
i=1Ni)×(Na+
∑N
i=1Ni) is the inertia
matrix for the arm and the fingers, M = diag (m,m,m)is the mass of the object, I = RIRT and I ∈ R
3×3 are
the inertia tensors for the object represented by the principal
axes of inertia. On the other hand, the total potential energy
for the overall system is given as follows:
P =
N∑
i=1
P (∆ri) =
N∑
i=1
∫ ri−∆ri
0
fi (∆ri) dζ, (13)
where P (∆ri) is the elastic potential energy for each finger
generated by the deformation of the finger tip. Hence,
Lagrange’s equation of motion is expressed by applying the
variational principle as follows:
For the multi-fingered hand-arm system:
H (q) q +
1
2H (q) + S (q, q)
q
+
N∑
i=1
(
JT0iCiY fi +XT
iqλiX +ZTiqλiZ
)
= u, (14)
For the object:
Mx+
N∑
i=1
(
−fiCiY +XTixλiX +ZT
ixλiZ
)
= 0 (15)
Iω +
1
2I + S
ω −
N∑
i=1
CiY × (x− x0i)fi
+
N∑
i=1
(
XTiωλiX +ZT
iωλiZ
)
= 0, (16)
where S (q, q) is skew-symmetric matrix, u is a vector of
the input torque. In addition, λiX and λiZ denote Lagrange’s
multipliers.
III. CONTROL INPUT
Tahara et al. [17] proposed a simple control method of a
triple robotic fingers system for stable grasping. We propose
a new control method based on Tahara’s method for a multi
fingered hand-arm system. This control signal is designed so
that the center of each finger tip approaches each other. The
control signal is given as follows:
u =fd
∑Ni=1 ri
N∑
j=1
J0j(xd − x0j)−Cq (17)
xd =1
N
N∑
i=1
x0i, (18)
where C ∈ R(Na+
∑N
i=1Ni)×(Na+
∑N
i=1Ni) > 0 is a diagonal
positive definite matrix that expresses the damping gain for
each finger, and fd is the nominal desired grasping force.
Now, an output vector of overall system is given as follows:
Λ = (q, x,ω)T. (19)
By substituting (17) into (14) and taking a sum of inner
product of (19) and closed loop dynamics expressed by (14),
(15) and (16), we obtain
d
dtE= −qTCq −
N∑
i=1
ξ∆r2i ≤ 0 (20)
E= K + V +∆P ≥ 0 (21)
K=1
2qTHq +
1
2xTMx+
1
2ωTIω (22)
V =A
2
(x01 − x02)2+ (x02 − x03)
2
+... +(x0N − x01)2
(23)
∆P =
N∑
i=1
∫ δri
0
fi (∆rdi + φ)− fi (∆rdi)
dφ, (24)
where
δri = ∆rdi −∆ri (25)
A =fd
N(
∑Ni=1 ri
) . (26)
∆rdi is an initial value of ∆ri. V plays a roll of an artificial
potential energy arisen from the control input. Equation (21)
is evident since K , V and ∆P are positive as far as 0 ≤∆rdi − δri < ri. In addition, E ≤ 0 during movement.
Equations (20) and (21) yield
∫
∞
0
(
qTCq +
N∑
i=1
ξ∆r2i
)
dt
≤ E (0)− E (t) ≤ E (0) , (27)
Equation (27) shows that the joint angular velocity q (t) is
squared integrable over time t ∈ [0,∞). It shows that q(t) ∈
2266
L2(0,∞). Considering the constraints shown by (4) and (5),
it is clear that x ∈ L2(0,∞) and ω ∈ L2(0,∞). Thereby,
the output of the overall system Λ(t) is uniformly continuous
since it is shown that Λ → 0 and Λ → 0 when t → ∞[15]. Therefore, it is obvious that the sum of the external
force applied to the finger system and the object ∆λ∞ is
converged to zero.
∆λ∞ = (∆λq,∆λx,∆λω) → 0 (28)
where
∆λq =
N∑
i=1
(
JT0iCiY fi +XT
iqλiX +ZTiqλiZ
)
− u (29)
∆λx=
N∑
i=1
(
−fiCiY +XTixλiX +ZT
ixλiZ
)
(30)
∆λω =−
N∑
i=1
CiY × (x− x0i)fi
+
N∑
i=1
(
XTiωλiX +ZT
iωλiZ
)
. (31)
As a consequence, it is shown the dynamic force/torque
equibilium condition for immobilization of the object is
satisfied since each external force and each velocity become
zero. In other words, the stability of the overall systems is
verified. However, it is necessary to consider the stability
from physical perspective since the stability is analyzed only
from a mathmatical viewpoint. There are some cases that one
of the centers of the contact areas is outside of the object
surfaces at the final state. Therefore, there is a requirement
to specify the shape of the object to argue the stability from
physical perspective. Moreover we argue a final state of the
overall system mainly, because an overall movement depends
on each damping gain and initial contact position but it is
not depended on the final state of the system. Thereby in
this paper, we focus on a final state of the overall system
as the initial step of convergence analysis. Of course, it is
important to consider the convergence of overall movement
and it is one of our next works.
In the next section, as an example, we introduce the
condition for an object with an arbitrary polygonal shape.
IV. SPECIFIC STUDY
In this section, the condition for stable grasping of an
arbitrary polyhedral object is shown. In addition, an example
of the condition for a triangle pole is introduced.
The final positions of the finger tips are obtained when
E is minimized. On the other hand, the positions of the
finger tips when E is minimized (= Emin) coincide with the
positions of the finger tips when V is minimized (= Vmin).Therefore, we derive the closed loop dynamics shown in (14)
to (16) and V in (23) for an arbitrary polyhedral object.
For the multi-fingered hand-arm system:
Hq+
1
2H + S +C
q
+N∑
i=1
(
NZiJT0i −∆riλiZ cos θsiJ
TΩi
)
rX
+
N∑
i=1
(
∆riλiXJTΩi − λiZJ
T0i
)
rY
+
N∑
i=1
(
−NZiJT0i −∆riλiZ sin θsiJ
TΩi
)
rZ
−A
N∑
i=1
N∑
j=1
NWj −NNWi
JT0i
rX
−A
N∑
i=1
N∑
j=1
Zj −NZi
JT0i
rY
−A
N∑
i=1
N∑
j=1
NWj −NNWi
JT0i
rZ
= 0, (32)
For the object:
Mx+
N∑
i=1
(
−NZirX + λiZrY +NZirZ
)
= 0 (33)
Iω+
1
2I + S
ω +
N∑
i=1
(
−NY iλiZ − NXiZi
)
rX
+
N∑
i=1
(fiXi + λiXYi) rY
+
N∑
i=1
(
−NY iλiZ +NXiZi
)
rZ = 0, (34)
where
θ1 = 0
θi = cos−1(
CT(i−1)Y CiY
)
(i = 2, 3, ..., N)
θsi =∑i
h=1 θhRCi = R−jθsiR−iπ
2
Di = Yi +∆riNWi = Xi sin θsi +Di cos θsiNWi = Xi cos θsi −Di sin θsiNXi = fi sin θsi + λiX cos θsiNXi = fi cos θsi − λiX sin θsiNY i = Yi cos θsi +Xi sin θsiNY i = Yi sin θsi −Xi cos θsiNZi = fi cos θsi + λiX sin θsiNZi = fi sin θsi − λiX cos θsi,
(35)
where Yi is the distance from the center of the object mass
Oc.m. to the surface. From (32), (33) and (34), the scalar
function V is given as follows:
2267
3X
1Y
z
x1X
3Y
2Y
y
x
x
y
1tθ
3θ
2X
x
y3tθ
2tθ
Fig. 3. Cross-section view of polygonal column
V = A[(
X21 +X2
2 + ...+X2N
)
+(
D21 +D2
2 + ...+D2N
)
−(D1X2 −X1D2) sin θt2 + (D2X3 −X2D3) sin θt3
+...+ (DNX1 −XND1) sin θt1
− (X1X2 +D1D2) cos θt2 + (X2X3 +D2D3) cos θt3
+...+ (XNX1 +DND1) cos θt1]
+A
2
(Z1 − Z2)2+ (Z2 − Z3)
2
+... +(ZN − Z1)2
, (36)
where
θti = θsi − θs(i−1) (i = 1) (37)
θti = θsi − θsN (i = 1). (38)
θti is the external angle of the polygon parallel to the base
of the object (see Fig. 3). One of the prerequisite for the
minimum value of V is
Z1 = Z2 = ... = ZN . (39)
In the case that the grasped object has arbitrary polyhedral
shape, Xi can be considered as an index for evaluating the
stability of the system from the physical perspective. In fact,
stable grasping is realized if Xi is inside of each contact
surface when V is minimized to Vmin, that is to say, the
final state of the system is stable in the physical viewpoint.
Equation (36) has three parameters Xi, Di and θti. As an
example, we show the relation between Xi and θti in Table I
and the relation between Xi and Di for Vmin in the case that
the object is a triangle pole, where the size of the triangle
pole is normalized by the cross-section area S. Table II shows
the relation between Xi and the radius of each finger tip rifor Vmin in order to show the relation between Xi and Di for
Vmin. In Table I and II, “IN” (see Fig. 4) means a case of
success in which all the centers of the contact areas are inside
of the object surface at the final state, and “OUT” (see Fig.
5) means a case of failure in which at least one of the centers
TABLE I
RELATIONSHIP BETWEEN θti AND Xi
θt1[rad] θt2[rad] θt3[rad] X1[m] X2[m] X3[m]
A 2.84 2.84 0.60 0.00 -0.0601 0.0601 INB 2.54 2.54 1.20 0.00 -0.0122 0.0122 INC 2.24 2.24 1.80 0.00 -0.0094 0.0094 IND 1.94 1.94 2.40 0.00 0.0190 -0.0190 INE 1.64 1.64 3.00 0.00 0.1137 -0.1137 OUT
(rj = 0.03 [m] ,∆rmin = 0.02 [m] , S = 6.38 × 103[
m2]
)
TABLE II
RELATIONSHIP BETWEEN ri AND Xi
r[m] ∆rmin[m] X1[m] X2[m] X3[m]
F 0.030 0.020 0.00 0.0504 -0.0504 ING 0.050 0.040 0.00 0.0605 -0.0605 INH 0.070 0.060 0.00 0.0705 -0.0705 OUT
(
(θt1, θt2, θt3) = (1.79, 1.79, 2.70) [rad], S = 6.38× 103[
m2])
C
A
F
Fig. 4. Sample case of “IN” as shown in Table I (A, C) and II (F). IN :The case of success in which all the centers of the contact areas are insideof the object surfaces at the final state (V = Vmin)
E H
Fig. 5. Sample case of “OUT” as shown in Table I (E) and II (H). OUT :The case of failure in which at least one of the centers of the contact areasis outside of the object surfaces at the final state (V = Vmin)
of the contact areas is outside of the object surfaces at the
final state. Table I and II show that the shape of the grasped
object and the radius of each finger tip determine whether a
stable grasping is realized or not at the final state. From these
considerations, we can say that initial positions of the finger
tips are valid for stable grasping if the initial positions are
close to the position obtained when V is minimized to Vmin.
Therefore, Vmin, the minimum value of the scalar function
V , can be used as an index for evaluating the possibility of
stable grasping.
V. NUMERICAL SIMULATION
We conduct numerical simulations for verifying the pro-
posed approach. The parameters of the triple-fingered hand-
arm system and the object in numerical simulation are shown
2268
TABLE III
PHYSICAL PARAMETERS
Triple-fingered hand-arm system
1st link length la1 1.300[m]
2nd link length la2 1.000[m]
3rd link length la3 0.175[m]
1st link length li1 0.300[m]
2nd link length li2 0.200[m]
1st mass ma1 1.300[kg] 1st mass center lga1 0.650[m]
2nd mass ma2 1.000[kg] 2nd mass center lga2 0.500[m]
3rd mass ma3 0.400[kg] 3rd mass center lga3 0.0875[m]
1st mass mi1 0.250[m] 1st mass center lgi1 0.150[m]
2nd mass mi2 0.150[m] 2nd mass center lgi2 0.100[m]
1st Inertia Ia1 diag(7.453, 7.453, 0.260)×10−1[kg·m2]2nd Inertia Ia2 diag(3.397, 3.397, 0.128)×10−1[kg·m2]3rd Inertia Ia3 diag(0.291, 0.291, 0.500)×10−1[kg·m2]1st Inertia Ii1 diag(7.725, 7.725, 0.450)×10−3[kg·m2]2nd Inertia Ii2 diag(2.060, 2.060, 0.120)×10−3[kg·m2]Radius of fingertip ri 0.070[m]
Stiffness coefficient ki 1.000×105[N/m2]Damping function ξi 1.000×
(
r2i −∆r2i)
π[Ns/m2]
Object
Mass m 0.037[kg]
(Y1, Y2, Y3) (0.092, 0.048, 0.048)[m]
(θt1, θt2, θt3) (1.833, 1.833, 2.618)[rad]
Inertia I diag (1.273, 0.193, 1.148) × 10−3[kg ·m2]
TABLE IV
DESIRED GRASPING FORCE AND GAINS
fd 1.0[N]Ca diag(3.080, 2.065, 2.483, 0.768, 0.487)×10−2[Ns·m/rad]
Ci diag(0.164, 0.177, 0.064)×10−2[Ns·m/rad]
TABLE V
INITIAL CONDITION
q 0[rad/s]
qa (−0.262,−1.571, 2.094, 0.785, 0.000)T [rad]
q01 (0.000,−0.663, 1.934)T × 10−2[rad]
q02 (−0.262,−0.705, 2.185)T × 10−2[rad]
q03 (0.262,−0.705, 2.185)T × 10−2[rad]x 0[m/s]
x (0.000, 0.600, 0.800)T [m]ω 0[rad/s]
R
1 0 00 1 00 0 1
in Table III. Table IV and V show the desired grasping
force and gains, and the initial condition, respectively. The
results of the grasping simulation are depicted in Figs. 6–
10. From Fig. 6 which indicates Xi and Zi, we can see
that Xi converges to which V satisfies Vmin, and thus, the
condition of (39) is satisfied. In addition, Xi and Zi converge
to the values which satisfy the conditions for Vmin. From
these results, we can conclude that the analysis of Vmin
is useful and it can be regarded as one of the index for
evaluating whether stable grasping is realized or not. This is
also effective for evaluating the initial positions of each finger
tips. Figures 7 and 8 show the elements of ∆λ∞ converges
to zero. It indicates that the sum of external force applied
to the system and the object converges to zero. The results
illustrate that the dynamic force/torque equibilium condition
for immobilization of the object is satisfied. Figures 9 and
0.05
Z
-0.01
0
0.01
É1
-0.09
-0.08
-0.07
-0.06
0 5 10 15
É1
fromV3
0.06
0.07
0.08
0.09
É1
fromV2
2X
min2 for VX
1X
min1 for VX
3X
min3 for VX
Xi
[m]
Time[s]
0.02
0.03
0.04
0 5 10 15
Z1
Z2
Z3
Time[s]
Zi
[m] 1Z
2Z
3Z
Fig. 6. History of Xi reach to Xi for Vmin and history of Zi satisfies(39)
-0.6
-0.3
0
0.3
0.6
0 5 10 15
àqf
àqf
àqf
ûq11
ûq13
ûq12
Time[s]
û
q[N
]1i
-1
-0.5
0
0.5
1
0 5 10 15
àqf
àqf
àqf
û
q[N
]2i
ûq21
ûq23
ûq22
Time[s]
-1
-0.5
0
0.5
1
0 5 10 15
àqf
àqf
àqf
ûq31
ûq33
ûq32
û
q[N
]3i
Time[s]
-3
-1.5
0
1.5
3
0 5 10 15
àqa
àqa
àqa
àqa
àqa
û
qa
[N]
ûqa1
ûqa3
ûqa4
ûqa2
ûqa5
Time[s]
Fig. 7. External force applied to the system convergence to zero
10 show q, x and ω and we can confirm that the velocities
of the overall system converge to zero.
VI. CONCLUSION
This paper presented the novel stable grasping method
for an arbitrary polyhedral object by a multi-fingered hand-
arm system. Firstly, the nonholonomic constraints of rolling
contact were formulated, and the conditions to realize the
stable grasping was derived from the stability analysis of
the overall system. Additionally, a new index obtained by
the scalar function Vmin was proposed for evaluating the
possibility of stable grasping. This possibility depends only
2269
-0.2
-0.1
0
0.1
0.2
0 5 10 15
à
à3
à2
û&1
û&3
û&2
Time[s]
û&
[N]
-2
0
2
4
0 5 10 15
àx3
àx2
àx1
ûx1
ûx3
ûx2
Time[s]
û
x[N
]
Fig. 8. External force applied to the object convergence to zero
-20
-10
0
10
20
0 5 10 15
theta_
theta_
theta_
11q&
12q&
13q&
An
gu
lar
Vel
oci
ty [
rad
/s]
Time[s]
-20
-10
0
10
20
0 5 10 15
theta_
theta_
theta_
21q&
22q&
23q&
Time[s]
An
gu
lar
Vel
oci
ty [
rad
/s]
-20
-10
0
10
20
0 5 10 15
theta_
theta_
theta_
31q&
32q&
33q&
Time[s]
-1
-0.5
0
0.5
1
0 5 10 15
theta_
theta_
theta_
theta_
theta_
Time[s]
1aq&
2aq&
3aq&
4aq&
5aq&
An
gu
lar
Vel
oci
ty [
rad
/s]
An
gu
lar
Vel
oci
ty [
rad
/s]
Fig. 9. History of joint angler velocity of arm and each finger
-0.1
-0.05
0
0.05
0.1
0 5 10 15
pos_v
pos_v
pos_v
-30
-15
0
15
30
0 5 10 15
ori_ve
ori_ve
ori_ve
Vel
oci
ty [
m/s
]
Time[s]
x&y&
z&
An
gu
lar
Vel
oci
ty [
rad
/s]
Time[s]
xω
yω
zω
Time[s]
Fig. 10. History of translational and rotational velocity of grasped object
on the shape of potential function V , and it does not
depend on initial contact positions and soft finger contact
model. The usefulness of the proposed index was verified
through numerical simulations. This means that the proposed
method realizes stable grasping regardless of any external
sensing and that the preshaping of the hand-arm system
for approaching to the object is important to realize stable
grasping.
Currently, the position and orientation of the object are not
specified explicitly. However, it would be easily possible to
control the position and orientation of the grasped object by
referring to the method proposed by Tahara et al. [17] and
Bae et al. [18]. In the future works, we would like to perform
some experiments to verify the usefulness of our proposed
method. Furthermore, we will expand the proposed control
scheme for an object with curved surfaces.
ACKNOWLEDGMENT
This work was partially supported by ”the Kyushu Univer-
sity Research Superstar Program (SSP)”, based on the budget
of Kyushu University allocated under President’s initiative.
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